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arXiv · 2111.00907

On a class of Hausdorff measure of cartesian product sets

Hajer Jebali·Riheb Guedri·Najmeddine Attia
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Abstract

In this paper, we study, in a separable metric space, a class of Hausdorff measures $\mathcal{H}_μ^{q, ξ}$ defined using a measure $μ$ and a premeasure $ξ$. We discuss a Hausdorff structure of product sets. Weighted Hausdorff measures $\mathcal{W}_μ^{q, ξ}$ appeared as an important tool when studying the product sets. When $μ$ and $ξ$ are blanketed, we prove that $\mathcal{H}_μ^{q, ξ} = \mathcal{W}_μ^{q, ξ}$. As an application, the case when $ξ$ is defined as the Hausdorff function is considered.

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Hajer Jebali, Riheb Guedri, Najmeddine Attia. 2024-01-06. On a class of Hausdorff measure of cartesian product sets. https://arxiv.org/abs/2111.00907

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